In: Statistics and Probability
The usable lifetime of a particular electronic component is known to follow an exponential distribution with a mean of 6.1 years. Let X = the usable lifetime of a randomly selected component. (a) The proportion of these components that have a usable lifetime between 5.9 and 8.1 years is __. (b) The probability that a randomly selected component will have a usable life more than 7.5 years is __. (c) The variance of X is __.
Given that the usable lifetime of a particular electronic component is known to follow an exponential distribution with a mean of 6.1 years.
Let,
X = the usable lifetime of a randomly selected component.
Before we go on to solve the problems let us know a bit about Exponential Distribution.
Exponential Distribution
A positive continuous random variable X is said to have an exponential distribution if its probability density function(PDF) is given by,
Notation:
Moments
Variance
Coming back to our problem,
In this problem,
X = the usable lifetime of a randomly selected component.
(a) Here we need to find the proportion of these components that have a usable lifetime between 5.9 and 8.1 years.
X = the usable lifetime of a randomly selected component.
(b) Here we need to find the probability that that a randomly selected component will have a usable life more than 7.5 years.
X = the usable lifetime of a randomly selected component.
Hence the probability that that a randomly selected component will have a usable life more than 7.5 years is 0.2924
(c) The variance of X is given by,
X = the usable lifetime of a randomly selected component.
We know that,
Hence the variance of X is 37.21